I did some research on the networking and server- client stuff, and I found that clients need to be connected to server by specifying ip address and port. Now this creeps me out - I also found that to use port number it is supposed to be registered in some global database that contains all of them, so I can't just pick random one and go with it. What I want to do is make a multi player component for my game, that might even use dedicated server (in my case not a real server - just separate pc). Seeing that there are many many programs and games using one of the ports and those have its range since they use maximum of five numbers, do I really need to register the port to use it in my game?

You don't need to register a port to use it. However, it is a good idea to consult the list of "known" or "common" ports, and pick a different one. This way, you may avoid collisions, should someone want to host your server software and the other server. It is a good idea to allow both the server and client to specify a different port.

While your server is listening on a specific port no other can do that and same thing around. If some other server process is already listening to a port you want to use you get an error saying something like "address already in use"

There is some kind of database available that can be querried for system defined ports and for what kind of service they are normaly used. But this is a different functionality.

This database is used to make applications idependent from fixed port numbers.

There is no enforced requirement. Nothing keeps you from setting up your game to run on port 80 even if it isn't using HTTP.

That usually does not cause problems but there are some firewalls out there which will manipulate anything going over port 80. Even if it is not actually HTTP. That will usually result in not getting any data or getting corrupted data. For a generally available program I would not advise using port 80.